An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 5.5

Elementary Processes of Quantum Electrodynamics — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · qed-massive-vector-boson

Problem

Add to QED a massive photon field BμB_\mu of mass MM, which couples to electrons via ΔH=∫d3x (gψˉγμψBμ).\Delta H = \int d^3x\, (g\bar{\psi}\gamma^\mu \psi B_\mu). A massive photon in the initial or final state has three possible physical polarizations, corresponding to the three spacelike unit vectors in the boson's rest frame, which can be characterized invariantly in terms of the boson's 4-momentum kμk^\mu, as the three vectors ϵμ(i)\epsilon_\mu^{(i)} satisfying ϵ(i)⋅ϵ(j)=−δij,k⋅ϵ(i)=0.\epsilon^{(i)} \cdot \epsilon^{(j)} = -\delta^{ij}, \qquad k \cdot \epsilon^{(i)} = 0. The four vectors (kμ/M,ϵμ(i))(k_\mu/M, \epsilon_\mu^{(i)}) form a complete orthonormal basis. Because BμB_\mu couples to the conserved current ψˉγμψ\bar{\psi}\gamma^\mu \psi, the Ward identity implies that kμk_\mu dotted into the amplitude for BB production gives zero; thus we can replace: ∑iϵμ(i)ϵν(i)∗  →  −gμν.\sum_i \epsilon_\mu^{(i)} \epsilon_\nu^{(i)*} \;\to\; -g_{\mu\nu}. This gives a generalization to massive bosons of the Feynman trick for photon polarization vectors and simplifies the calculation of BB production cross sections. (Warning: This trick does not work (so simply) for "non-Abelian gauge fields".) Let's do a few of these computations, using always the approximation of ignoring the mass of the electron. (a) Compute the cross section for the process e+e−→Be^+e^- \to B. Compute the lifetime of the BB, assuming that it decays only to electrons. Verify the relation σ(e+e−→B)=12π2MΓ(B→e+e−)δ(M2−s)\sigma(e^+e^- \to B) = \frac{12\pi^2}{M} \Gamma(B \to e^+e^-) \delta(M^2 - s) discussed in Section 5.3. (b) Compute the differential cross section, in the center-of-mass system, for the process e+e−→γ+Be^+e^- \to \gamma + B. (This calculation goes over almost unchanged to the realistic process e+e−→γ+Z0e^+e^- \to \gamma + Z^0; this allows one to measure the number of decays of the Z0Z^0 into unobserved final states, which is in turn proportional to the number of neutrino species.) (c) Notice that the cross section of part (b) diverges as θ→0\theta \to 0 or π\pi. Let us analyze the region near θ=0\theta = 0. In this region, the dominant contribution comes from the tt-channel diagram and corresponds intuitively to the emission of a photon from the electron line before e+e−e^+e^- annihilation into a BB. Let us rearrange the formula in such a way as to support this interpretation. First, note that the divergence as θ→0\theta \to 0 is cut off by the electron mass: Let the electron momentum be pμ=(E,0,0,k)p^\mu = (E, 0, 0, k), with k=(E2−me2)1/2k = (E^2 - m_e^2)^{1/2}, and let the photon momentum be kμ=(xE,xEsin⁡θ,0,xEcos⁡θ)k^\mu = (xE, xE\sin\theta, 0, xE\cos\theta). Show that the denominator of the propagator then never becomes smaller than O(me2/s)\mathcal{O}(m_e^2/s). Now integrate the cross section of part (b) over forward angles, cutting off the θ\theta integral at θ2∼(me2/s)\theta^2 \sim (m_e^2/s) and keeping only the leading logarithmic term, proportional to log⁡(s/me2)\log(s/m_e^2). Show that, in this approximation, the cross section for forward photon emission can be written σ(e+e−→γ+B)≈∫dx f(x)⋅σ(e+e−→B at Ecm2=(1−x)s),\sigma(e^+e^- \to \gamma + B) \approx \int dx\, f(x) \cdot \sigma(e^+e^- \to B \text{ at } E_{\text{cm}}^2 = (1-x)s), where f(x)f(x) is a function that can be interpreted as the probability of radiating a photon with fractional momentum xx from the initial electron. Give the explicit formula for f(x)f(x).

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